What is \tan^2 \alpha equal to?
Consider the following for the three (03) items that follow : Let p=\tan 2\alpha-\tan \alpha and q=\cot \alpha-\cot 2\alpha.
- A. (pq)/(p+q)
- B. (p+2q)/p
- C. p/(p+2q) ✓
- D. p/(2p+q)
Correct Answer: C. p/(p+2q)
Explanation
From the first question, p/q = \tan \alpha \tan 2\alpha = \tan \alpha \left(\frac{2\tan \alpha}{1-\tan^2 \alpha}\right) = \frac{2\tan^2 \alpha}{1-\tan^2 \alpha}. Let x = \tan^2 \alpha. Then p/q = \frac{2x}{1-x} \implies p - px = 2qx \implies p = x(p+2q) \implies x = \frac{p}{p+2q}.